Primitive sets with large counting functions

A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\H os implies that $\int_2^\infty (S(t)/t^2\log t) dt$ converges. We establish an approximate converse to this theorem, showing that if $F$ satisfies some mild conditions and $\int_2^\infty (F(t)/t^2\log t) dt$ converges, then there exists a primitive set $S$ with $S(x) \gg F(x)$.

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